Emergence of dark phases in scalar particles within the Schwarzschild–Kiselev–Letelier spacetime

Abstract This work focuses on the emergence of dark phases (dark energy-induced phases) in the radial wave function of scalar particles. We achieve this by presenting novel solutions to the Klein–Gordon equation in a spherically symmetric spacetime, which encompasses a black hole, a quintessential fluid, and a cloud of strings. We determine the exact solution for the spacetime metric, analyze the admissible ranges for its physical parameters, and discuss the formation of the event horizon. Subsequently, we detail the solution of the Klein–Gordon equation and explore three distinct cases of dark phases, corresponding to the quintessence state parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">Q</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> taking the values 0, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , and 1. Notably, the case where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mrow> <mml:mrow> <mml:mstyle mathsize="0.5em"> <mml:mrow> <mml:mi mathvariant="normal">Q</mml:mi> </mml:mrow> </mml:mstyle> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> holds particular significance due to current observational constraints on dark energy.